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Topics in Applied Mathematics

Mathematics was coined the “queen of sciences” by the “prince of mathematicians,” Carl Friedrich Gauss, one of the greatest mathematicians of all time. Indeed, the name of Gauss is associated with essentially all areas of mathematics, and in this respect, there is really no clear boundary between “pure mathematics” and “applied mathematics.” To ensure financial independence, Gauss decided on a stable career in astronomy, which is one of the oldest sciences and was perhaps the most popular one...

Topics: Architecture, General Engineering, Actuarial Science, Algebra, Differential Equations, General Mathematics, Linear Algebra, Astronomy, Physics, Education
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Mathematics was coined the “queen of sciences” by the “prince of mathematicians,” Carl Friedrich Gauss, one of the greatest mathematicians of all time. Indeed, the name of Gauss is associated with essentially all areas of mathematics, and in this respect, there is really no clear boundary between “pure mathematics” and “applied mathematics.” To ensure financial independence, Gauss decided on a stable career in astronomy, which is one of the oldest sciences and was perhaps the most popular one during the eighteenth and nineteenth centuries. In his study of celestial motion and orbits and a diversity of disciplines later in his career, including (in chronological order): geodesy, magnetism, dioptrics, and actuarial science, Gauss has developed a vast volume of mathematical methods and tools that are still instrumental to our current study of applied mathematics.During the twentieth century, with the exciting development of quantum field theory, with the prosperity of the aviation industry, and with the bullish activity in financial market trading, and so forth, the sovereignty of the “queen of sciences” has turned her attention to the theoretical development and numerical solutions of partial differential equations (PDEs). Indeed, the non-relativistic modeling of quantum mechanics is described by the Schrödinger equation; the fluid flow formulation, as an extension of Newtonian physics by incorporating motion and stress, is modeled by the Navier-Stokes equation; and option stock trading with minimum risk can be modeled by the Black-Scholes equation. All of these equations are PDEs. In general, PDEs are used to describe a wide variety of phenomena, including: heat diffusion, sound wave propagation, electromagnetic wave radiation, vibration, electrostatics, electrodynamics, fluid flow, and elasticity, just to name a few. For this reason, the theoretical and numerical development of PDEs has been considered the core of applied mathematics, at least in the academic environment.On the other hand, over the past decade, we have been facing a rapidly increasing volume of “information” contents to be processed and understood. For instance, the popularity and significant impact of the open education movement (OEM) has contributed to an enormous amount of educational information on the web that is difficult to sort out, due to unavoidable redundancy, occasional contradiction, extreme variation in quality, and even erroneous opinions. This motivated the founding of the “Saylor Foundation courseware” to provide perhaps one of the most valuable, and certainly more reliable, high-quality educational materials, with end-to-end solutions, that are free to all. With the recent advances of various high-tech fields and the popularity of social networking, the trend of exponential growth of easily accessible information is certainly going to continue well into the twenty-first century, and the bottleneck created by this information explosion will definitely require innovative solutions from the scientific and engineering communities, particularly those technologists with better understanding of and a strong background in applied mathematics. In this regard, mathematics extends its influence and impact by providing innovative theory, methods, and algorithms to virtually every discipline, far beyond sciences and engineering, for processing, transmitting, receiving, understanding, and visualizing data sets, which could be very large or live in some high-dimensional space.Of course the basic mathematical tools, such as PDE methods and least-squares approximation introduced by Gauss, are always among the core of the mathematical toolbox for applied mathematics. But other innovations and methods must be integrated in this toolbox as well. One of the most essential ideas is the notion of “frequency” of the data information. Joseph Fourier, a contemporary of Gauss, instilled this important concept to our study of physical phenomena by his innovation of trigonometric series representations, along with powerful mathematical theory and methods, to significantly expand the core of the toolbox for applied mathematics. The frequency content of a given data set facilitates the processing and understanding of the data information. Another important idea is the “multi-scale” structure of data sets. Less than three decades ago, with the birth of another exciting mathematical subject, called “wavelets,” the data set of information can be put in the wavelet domain for multi-scale processing as well. On the other hand, it is unfortunate that some essential basic mathematical tools for information processing are not commonly taught in a regular applied mathematics course in the university. Among the commonly missing ones, the topics that will be addressed in this Saylor course MA304 include: information coding, data dimensionality reduction, data compression, and image manipulation.The objective of this course is to study the basic theory and methods in the toolbox of the core of applied mathematics, with a central scheme that addresses “information processing” and with an emphasis on manipulation of digital image data. Linear algebra in the Saylor Foundation’s MA211 and MA212 are extended to “linear analysis” with applications to principal component analysis (PCA) and data dimensionality reduction (DDR). For data compression, the notion of entropy is introduced to quantify coding efficiency as governed by Shannon’s Noiseless Coding theorem. Discrete Fourier transform (DFT) followed by an efficient computational algorithm, called fast Fourier transform (FFT), as well as a real-valued version of the DFT, called discrete cosine transform (DCT) are discussed, with application to extracting frequency content of the given discrete data set that facilitates reduction of the entropy and thus significant improvement of the coding efficiency. DFT can be viewed as a discrete version of the Fourier series, which will be studied in some depth, with emphasis on orthogonal projection, the property of positive approximate identity of Fejer’s kernels, Parseval’s identity and the concept of completeness. The integral version of the sequence of Fourier coefficients is called the Fourier transform (FT). Analogous to the Fourier series, the formulation of the inverse Fourier transform (IFT) is derived by applying the Gaussian function as a sliding time-window for simultaneous time-frequency localization, with optimality guaranteed by the Uncertainty Principle. Local time-frequency basis functions are also introduced in this course by discretization of the frequency-modulated sliding time-window function at the integer lattice points. Replacing the frequency modulation by modulation with the cosines avoids the Balian-Low stability restriction on the local time-frequency basis functions, with application to elimination of blocky artifact caused by quantization of tiled DCT in image compression. Gaussian convolution filtering also provides the solution of the heat (partial differential) equation with the real-line as the spatial domain. When this spatial domain is replaced by a bounded interval, the method of separation of variables is applied to separate the PDE into two ordinary differential equations (ODEs). Furthermore, when the two end-points of the interval are insulated from heat loss, solution of the spatial ODE is achieved by finding the eigenvalue and eigenvector pairs, with the same eigenvalues to govern the exponential rate of decay of the solution of the time ODE. Superposition of the products of the spatial and time solutions over all eigenvalues solves the heat PDE, when the Fourier coefficients of the initial heat content are used as the coefficients of the terms of the superposition. This method is extended to the two-dimensional rectangular spatial domain, with application to image noise reduction. The method of separation of variables is also applied to solving other typical linear PDEs. Finally, multi-scale data analysis is introduced and compared with the Fourier frequency approach, and the architecture of multiresolution analysis (MRA) is applied to the construction of wavelets and formulation of the multi-scale wavelet decomposition and reconstruction algorithms. The lifting scheme is also introduced to reduce the computational complexity of these algorithms, with applications to digital image manipulation for such tasks as progressive transmission, image edge extraction, and image enhancement.

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  • Institution: Saylor
  • Topics: Architecture, General Engineering, Actuarial Science, Algebra, Differential Equations, General Mathematics, Linear Algebra, Astronomy, Physics, Education

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About Saylor: The mission of the Saylor Foundation is to make education freely available to all. Guided by the belief that technology has the potential to circumvent barriers that prevent many individuals from participating in traditional schooling models, the Foundation is committed to developing and advancing inventive and effective ways of harnessing technology in order to drive the cost of education down to zero

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